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A113431
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Expansion of f(-x) * f(-x^10) / f(-x^4, -x^6) in powers of x where f() is Ramanujan's two variable theta function.
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0
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1, -1, -1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
This is an example of the quintuple product identity in the form f(a*b^4, a^2/b) - (a/b) * f(a^4*b, b^2/a) = f(-a*b, -a^2*b^2) * f(-a/b, -b^2) / f(a, b) where a = x^3, b = x^2.
f(a,b) = Sum_{k} a^((k^2+k)/2)*b^((k^2-k)/2) is Ramanujan's two-variable theta function.
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LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Eric Weisstein's World of Mathematics, Quintuple Product Identity
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FORMULA
| Expansion of f(x^4, x^11) - x * f(x, x^14) = f(-x^5, -x^10) * f(-x, -x^4) / f(x^2, x^3) in powers of x where f() is Ramanujan's two variable theta function.
Expansion of H(x^2) * f(-x) where H() is g.f. of A003106.
Euler transform of period 10 sequence [ -1, -1, -1, 0, -1, 0, -1, -1, -1, -1, ...].
G.f.: Sum_{k} x^(5*k * (3*k + 1) / 2) * (x^(-6*k) - x^(6*k + 2)).
G.f.: Prod_{k>0} (1 - x^k) / ((1 - x^(10*k - 4)) * (1 - x^(10*k - 6))).
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EXAMPLE
| 1 - x - x^2 + x^4 + x^11 - x^15 - x^18 + x^23 + x^37 - x^44 - x^49 + ...
q^49 - q^169 - q^289 + q^529 + q^1369 - q^1849 - q^2209 + q^2809 + q^4489 + ...
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PROG
| (PARI) {a(n) = local(m); if( n<0 || !issquare( n*120 + 49, &m) || 1! = gcd(m, 30), 0, (-1)^( m%30 \ 10))}
(PARI) {a(n) = if( n<0, 0, polcoeff( prod( k=1, n, 1 - x^k*[ 1, 1, 1, 1, 0, 1, 0, 1, 1, 1][k%10 + 1], 1 + x * O(x^n)), n))}
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CROSSREFS
| Cf. A003106.
Sequence in context: A071003 A071002 * A116915 A076141 A011751 A093709
Adjacent sequences: A113428 A113429 A113430 * A113432 A113433 A113434
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KEYWORD
| sign,changed
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AUTHOR
| Michael Somos, Oct 31 2005
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